now :: thesis: ( not 2 divides 439 & not 3 divides 439 & not 5 divides 439 & not 7 divides 439 & not 11 divides 439 & not 13 divides 439 & not 17 divides 439 & not 19 divides 439 )

439 = (2 * 219) + 1 ;
hence not 2 divides 439 by NAT_4:9; :: thesis: ( not 3 divides 439 & not 5 divides 439 & not 7 divides 439 & not 11 divides 439 & not 13 divides 439 & not 17 divides 439 & not 19 divides 439 )
439 = (3 * 146) + 1 ;
hence not 3 divides 439 by NAT_4:9; :: thesis: ( not 5 divides 439 & not 7 divides 439 & not 11 divides 439 & not 13 divides 439 & not 17 divides 439 & not 19 divides 439 )
439 = (5 * 87) + 4 ;
hence not 5 divides 439 by NAT_4:9; :: thesis: ( not 7 divides 439 & not 11 divides 439 & not 13 divides 439 & not 17 divides 439 & not 19 divides 439 )
439 = (7 * 62) + 5 ;
hence not 7 divides 439 by NAT_4:9; :: thesis: ( not 11 divides 439 & not 13 divides 439 & not 17 divides 439 & not 19 divides 439 )
439 = (11 * 39) + 10 ;
hence not 11 divides 439 by NAT_4:9; :: thesis: ( not 13 divides 439 & not 17 divides 439 & not 19 divides 439 )
439 = (13 * 33) + 10 ;
hence not 13 divides 439 by NAT_4:9; :: thesis: ( not 17 divides 439 & not 19 divides 439 )
439 = (17 * 25) + 14 ;
hence not 17 divides 439 by NAT_4:9; :: thesis: not 19 divides 439
439 = (19 * 23) + 2 ;
hence not 19 divides 439 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 439 & n is prime holds
not n divides 439 by XPRIMET1:16;
hence 439 is prime by NAT_4:14; :: thesis: verum